is any upper bound for mean square error of an unbiased estimator?

Question

There is always a lower bound for an unbiased estimator called Cramer-Rao Lower Bound. Does any one remember any upper bound for unbiased estimator? The upper bound is used for worst-case analysis of my estimator performance in telecommunication engineering.

Answer 1

Such a bound does not make sense. Take $x(t)=A+B\sin(2\pi f_0 t)$ and let your estimator be $\Theta{'}(t)=x(t)$ for some uniformly distributed $t\in[0,T]$. Then, the expected value of your estimator $\Theta{'}(t)$ can be made the same with the true parameter $\Theta$ through increasing (or decreasing) $A$. Hence, this estimator will be unbiased. When $B\rightarrow \infty$, The variance will also go to infinity. As a result one cannot talk about any upperbound of an unbiased estimator.

If additionally the estimator should be minimum variance estimator (MVUnbiasedE), then this estimator is unique therefore such a comparison can also not be done among MVUEs, because there is only one.